On the conditioning of robustness problems

The focal point of this paper is the \´\´conditioning\´\´ of robustness problems with performance specifications depending nonlinearly on uncertain parameters. Beginning with a rather general class of such problems, we define a so-called underlying conditioner Φ. For large classes of robustness problems characterized by the requirement that a nonlinear function f(x) be negative on a prescribed set X in R_n, it is seen that the associated conditioner Φ serves as a natural measure of the degree of difficulty which one might expect to encounter in order to certify that the volume of violation in parameter space is below some acceptable level. In this paper, a number of properties of the conditioner Φ are described. Most notably, it is shown how this conditioner can be estimated "on the fly" within the context of a robustness computation. For the large class of robustness problems with f(x) being a multivariable polynomial and X being a hypercube, a convergent sequence of estimates for Φ is obtained.